Fourier series and Integrals

Question

Let $F_N$ be Fejer kernal,namely $F_N=\dfrac{1}{N+1}(\dfrac{sin(\pi(N+1)t))}{sin(\pi t)})^2$.If $f\in L^p$,has period $1$,$\sigma_N f(x)=\int_0^1f(x-t)F_N(t)dt$, my question is why the following equation is ture? $$\lVert \sigma_Nf-f\rVert_p=\int_{-1/2}^{1/2}\lVert f(x-t)-f(x)\rVert_pF_N(t)dt$$ By my calculation,I believe that $$\lVert \sigma_Nf-f\rVert_p=\lVert\int_{-1/2}^{1/2}(f(x-t)-f(x))F_N(t)dt\rVert_p$$